Finite Mathematics Lesson 12

 Section 10.1 Section 10.2 Section 10.3

Chapters 10.1, 10.2, 10.3

You are almost finished!  The last sections have a personal application as well as a business application.  Most of us are paying off debt as well as trying to save money for the future. Do you really know how much of a mortgage payment actually goes toward the principle?  Is the investment vehicle for your retirement going to allow you to retire before the age of 50?  Chapter 10 will look into these type of questions.  The TI-83 has an feature called the TVM solver which is an excellent tool.  Check it out the online calculators in the course content.  Make sure you can show your work by hand on the final.

Course Notes

Section 10.1    Interest

Read 10.1 and review the examples.  These problems should be a review for you from College Algebra.  The formulas may look a little different than in most books but the concepts are still the same.  Make sure you can make the distinction between present value and future value.

Important Formulas to Remember

 Compound Interest Present Value Simple Interest A = (1 + nr)P
 Example 1: You want to invest \$20,000 for 30 years  at 11 % interest compounded quarterly.  How much money will you have at the end of the 30 years?  (before taxes) Use the compound interest formula with... i = .11/4 = 0.0275  quarterly means 4 times a year (Hint: don't round until you are completely finished with your calculations) P = present Value = 20,000 , n = number of interest period = 30 × 4 = 120  and we want to find F = Future value F = (1 + 0.0275)120 × 20,000 = 25.93102392 × 20,000 = 518,620.48 Example 2: Let's say you want to retire in 30 years with a million dollars.  You invest some money in a mutual fund that expects to earn an average of 12% per year compounded monthly.  How much money do you need to invest? Use the present value formula... i = .12/12 = 0.01  monthly means 12 times a year F = Future value = 1,000,000  and n = number of interest period = 30 × 12 = 360 and we want to find P = Present value P = ( 1÷ (1 + .01)360 ) × 1,000,000 = 0.0278166892 × 1,000,000 = 27,816.69 Be careful of rounding errors.  Do not round until the after last calculation!!!! Example 3: You decide to invest in a business deal that pay simple interest.   Calculate the amount after 30 years if \$20,000 is deposited at 11% simple interest. Use the simple interest formula A = P + nrP = (1 + nr)P i = .11 , n = 30 and P = 20,000 A = (1 + 30 × .11) × 20,000 = 4.3 × 20,000 = 86,000 Note: This amount is considerably less than the amount in example 1.

Section 10.2    Annuities

• An Annuity is a sequence of equal payments made at regular intervals of time.

Example 2 (page 466 illustration 1)
As the proud parent of a newborn daughter, you decide to save for her college education by depositing \$100 at the end of each month into a savings account paying 6% interest compounded monthly.  Eighteen years from now, after you make the last of 216 payments, the account will contain \$38,735.32.

 The \$100 that is deposited every month will be called Rent and denoted as R. The interest i = rate ÷ times per year = .06 ÷ 12 = .005  or ½ % The number of payments n = 18 years × 12 months = 216

The formula for the Future Value of an Annuity is derived on page 436.

 Also written as I can't write it exactly as the book does. This part will be used in other equations and is easier to write that the entire expression.

Put it all together...

Notice  Make sure you extend you decimal places out.  If you are not sure how many decimal place, don't round until the final step.

A variation on example 1

You are still the proud parent of a newborn daughter, but you find that your daughter's college education will cost \$100,000 .  How much should you put away each month into a savings account paying 6% interest compounded monthly so that eighteen years from now, the account will contain \$100,000 ?

 We need to find the amount deposited every month called Rent and denoted as R. The interest i = rate ÷ times per year = .06 ÷ 12 = .005  or ½ % The number of payments n = 18 years × 12 months = 216

Solving the previous formula for R, we get....

 plugging this into our formula You would have to put away \$258.16 every month for 18 years.

Example 2
Having just won the state lottery, you decide not to work for the next 5 years.   You want to deposit enough money into the bank so that you can withdraw \$5,000 at the end of each month for 60 months.  If the banks pays 6% interest compounded monthly, you must deposit \$258,627.80.

This is a Present Value of an Annuity problem.
The amount of Rent, \$5,000 (the word rent can mean money received as well as money paid), is known.  We want to find the amount of money, needed to in the account  to allow us to withdraw \$5,000 a month.

 We need to find the amount deposited  called Present Value and denoted as P. The amount of Rent R = \$5,000 The interest i = rate ÷ times per year = .06 ÷ 12 = .005  or ½ % The number of payments n = 5 years × 12 months = 60
 We need  new formulas... Goofy notation, derived on page 439

P = 51.725555 × R = 51.725555 × 5,000 = 258,627.80

You must deposit \$258,627.80 today in order to withdraw \$5,000 a month for 5 years.

A variation on example 2
Having just won \$1,000,000 after taxes in the state lottery,  you want to know how much money ,Rent,  you can withdraw at the end of each month for 20 years.   If the banks pays 10% interest compounded quarterly, how much should you deposit?

 We need to find the amount deposited called Present Value and denoted as P. The amount of Rent R is unknown. Important Remark   The frequency of payment and the interval of compounding must be in agreement.  Since the interest will be compounded quarterly, we will calculate the Rent quarterly.  Then divide this value by 4 to get the monthly rent. The interest i = rate ÷ times per year = .10 ÷ 4 = .025 The number of payments n = 20 years × 4 = 80

 R = .029026045 × 1,000,000 = 29,026.05 This is the quarterly rent. 29,026.05 ÷ 4 = 7,256.51 Divide by 4 to get monthly amount.

You must deposit \$1,000,000 today in order to withdraw \$7,256.51 a month for 20 years.

Example 3
Is it more profitable to receive a lump sum of \$10,000 at the end of 3 years or to receive \$750 at the end of each quarter-year for 3 years?  Assuming that money can earn 8% interest compounded quarterly.

Notice that you will receive the lump sum at the end of 3 years.  Hence,  no chance to earn interest on that money.  Is this a Future Value question or a Present Value question?  Since the goal is to find out how much we have at the end of the 3 years this is a Future Value question.

 All we need to find is the Future Value of the money.  That is how much money will be paid out. The amount of Rent R = 750 The interest i = rate ÷ times per year = .08 ÷ 4 = .02 The number of payments n = 3 years × 4 = 12

Make sure you pick the right formula.

 We need Future Value and this....

F = 13.41209 × 750 = 10,059.07

You are better off taking \$750 per quarter.  Note:   \$750 × 4 times a year × 3 years = \$9,000 is the actual amount rent paid, the remaining \$1059.07 is the interest earn.

Section 10.3   Amortization of Loans

All of us at some point will have an amortized loan.  Each time a payment is made some of the money goes toward principle and the rest toward the interest. This sequence of payment constitutes an annuity.  The loans that are considered in this section

• will be repaid in a sequence equal payments
• at regular time intervals
• payments coincide with interest periods
• at the end of the interest periods

The sequence of payments constitutes an annuity so we can use the tools developed in 10.2

Example 4
James buys a house for \$90,000.  He puts \$10,000 down and then finances the rest at 9% interest compounded monthly for 25 years.

1. Find his monthly payments
This is a Present Value of an Annuity problem. We need to find the Rent R
 Note: n = 12(25) = 300 , i = .09/12 = .0075 and P = 90,000 - 10,000 = 80,000 The monthly payments are \$671.36
2. Find the total amounts he pays for the house.
You pay \$10,000 down and \$671.36 a month for 25 years
10,000 + 671.36 × 12 × 25 = 10,000 + 201,408 = 211,408
You pay \$211,408 for the house with interest.
3. Find the total amount of interest he pays.
211,408 - 90,000 = \$121,408
4. Find the amount he still owes after 23 years.
We need to find the Present Value of an Annuity using...
 P = 21.889146 × 671.36 P = \$14,695.50

Where did we get the 12 × 2 =24   from?
We used  n = 12 × 25 = 300 to find out out the Rent in part a.  In order to find the amount still owed after 23 years, figure you how many years are remaining on the loan. 25 - 23 = 2 years.  In this case n = 12 × 2 = 24

5. Find the amount he still owes after 24 years.
Same as part d except n = 12 × 1 = 12 since there is 1 year remaining on the loan.
 P = 11.434913 × 671.36 P = \$7676.94
6. Find the total amount of interest he pays in year 24.
After 1 hour of struggling with this problem, I do know how you feel, I decided to make an Excel chart..  I hid some rows for size considerations.  Try this in Excel. It works!

I found the solution in the solutions manual to be in error!
We need the difference of the amount owed at the end of year 23  and the amount owed at the end of year 24.
As we calculated  above, \$14,695.50 - P = \$7676.94 = \$7,018.56   This is the amount of the loan paid off in year 24.

Now, there are 12 payments in year 24.
12(671.36) = \$8,056.32  amount paid (including interest) in year 24.

Subtract the amount paid in year 24 from the amount of the loan paid off gives...
\$8,056.32 - \$7,018.56 = \$1,037.76    which is the amount Excel calculated.

7. Prepare a amortization table for the first 3 months.
I will use the same formulas used to make the excel chart, but by hand.
 Payment Amount Interest Applied to Principal Unpaid Balance 0 .09/12 = .0075 \$80,000 Steps Fixed = \$671.36 .0075*Unpaid Balance Subtract interest from Payment Subtract Principal from Unpaid Balance 1 \$671.36 .0075(80,000)= \$600 671.36 - 600= \$71.36 80,000 - 71.36 = \$79,928.64 2 \$671.36 .0075(79,928.64)= \$599.47 671.36 - 599.47= \$71.90 79,928.64 - 71.90 = \$79,856.74 3 \$671.36 .0075(79,856.74)= \$598.93 671.36 - 598.93= \$72.43 79,928.64 - 72.43 = \$79,784.31

We used this formula below Bnew is the Unpaid Balance and of course R is the rent 671.36.
Note Bnew = (1 + i )Bprevious - R  =     (1 + .0075)80,000 - 671.36 = 79,928.64

 If \$P is borrowed at interest rate i per period and \$R is paid back at the end of each interest period, then the formula Bnew = (1 + i )Bprevious - R  can be used to calculate each new balance, Bnew ,from the previous balance.